On~the number of zeros of the function $\zeta(s)$ on ``almost all'' short intervals of the critical line

L. V. Kiseleva

Izvestiya. Mathematics , Tome 32 (1989) no. 3, pp. 475-499

Résumé

Suppose $\varepsilon>0$ is an arbitrarily small fixed number, $$ Y\geqslant Y_0(\varepsilon)>0,\quad H=Y^\varepsilon,\quad Y_1=Y^{\frac{11}{12}+\varepsilon},\quad Y\leqslant T\leqslant Y+Y_1. $$ Consider the relation $$ N_0(T+H)-N_0(T)\geqslant cH\ln T, $$ where $c=c(\varepsilon)>0$ is a constant depending only on $\varepsilon$, and let $E$ denote the set of those $T$ in the interval $Y\leqslant T\leqslant Y+Y_1$ for which this relation does not hold. It is shown that the measure of this set satisfies $\mu(E)\leqslant Y_1Y^{-0.5\,\varepsilon}$. Bibliography: 19 titles.